Simplify the following expression: $\dfrac{84a}{120a^5}$ You can assume $a \neq 0$.
Solution: $ \dfrac{84a}{120a^5} = \dfrac{84}{120} \cdot \dfrac{a}{a^5} $ To simplify $\frac{84}{120}$ , find the greatest common factor (GCD) of $84$ and $120$ $84 = 2 \cdot 2 \cdot 3 \cdot 7$ $120 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$ $ \mbox{GCD}(84, 120) = 2 \cdot 2 \cdot 3 = 12 $ $ \dfrac{84}{120} \cdot \dfrac{a}{a^5} = \dfrac{12 \cdot 7}{12 \cdot 10} \cdot \dfrac{a}{a^5} $ $\phantom{ \dfrac{84}{120} \cdot \dfrac{1}{5}} = \dfrac{7}{10} \cdot \dfrac{a}{a^5} $ $ \dfrac{a}{a^5} = \dfrac{a}{a \cdot a \cdot a \cdot a \cdot a} = \dfrac{1}{a^4} $ $ \dfrac{7}{10} \cdot \dfrac{1}{a^4} = \dfrac{7}{10a^4} $